Subgroups — C₁₀₉×#20365 #42138359

Order 27904.

1054 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₁₀₉	C₂	1	2	yes	maximal
2	C₂×#376×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1746	C₂	1	8	yes	maximal, complemented
4	C₁₀₉×#1728	C₂	1	4	yes	maximal
5	C₂²×M₁₆×C₁₀₉	C₂²	1	4	yes	—
6	C₂²×C₄²×C₁₀₉	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₄×M₁₆×C₁₀₉	C₄	1	16	yes	complemented
9	C₂×#97×C₁₀₉	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₁₀₉	C₂²	1	4	yes	—
11	#376×C₁₀₉	C₄	1	4	yes	—
12	#376×C₁₀₉	C₂²	1	4	yes	—
13	C₂×#95×C₁₀₉	C₄	1	4	yes	—
14	#336×C₁₀₉	C₂²	1	16	yes	—
15	C₂³×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
16	C₂³×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₁₀₉	—	2	16	no	—
19	C₂×M₁₆×C₁₀₉	C₂×C₄	1	16	yes	—
20	C₂²×C₈×C₁₀₉	C₂³	1	8	yes	—
21	C₂×C₄²×C₁₀₉	—	2	8	no	—
22	C₂×C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
23	C₂×C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₀₉	C₂×C₄	1	8	yes	—
25	C₂×C₄²×C₁₀₉	C₂³	1	4	yes	—
26	C₂×C₄²×C₁₀₉	C₂³	1	2	yes	—
27	#97×C₁₀₉	C₂×C₄	1	16	yes	complemented
28	C₄×C₈×C₁₀₉	C₂×C₄	1	16	yes	complemented
29	#95×C₁₀₉	C₂×C₄	1	16	yes	—
30	C₂⁴×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	—	2	8	no	—
32	C₂²×C₄×C₁₀₉	—	4	4	no	—
33	C₂²×C₄×C₁₀₉	—	4	8	no	—
34	C₂²×C₄×C₁₀₉	C₄²	1	8	yes	—
35	C₂²×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
36	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
37	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
39	C₂²×C₄×C₁₀₉	#41	1	8	yes	—
40	C₂²×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
41	M₁₆×C₁₀₉	—	2	32	no	—
42	C₂×C₈×C₁₀₉	—	2	16	no	—
43	C₂×C₈×C₁₀₉	C₂²×C₄	1	16	yes	—
44	C₄²×C₁₀₉	—	2	8	no	—
45	C₄²×C₁₀₉	—	2	8	no	—
46	C₄²×C₁₀₉	C₄²	1	8	yes	—
47	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
48	C₄²×C₁₀₉	C₂²×C₄	1	4	yes	—
49	C₂³×C₁₀₉	—	2	4	no	—
50	C₂³×C₁₀₉	C₂×C₄²	1	2	yes	—
51	C₂³×C₁₀₉	#117	1	4	yes	—
52	C₂³×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
53	C₂×C₄×C₁₀₉	—	2	8	no	—
54	C₂×C₄×C₁₀₉	—	2	8	no	—
55	C₂×C₄×C₁₀₉	—	2	16	no	—
56	C₂×C₄×C₁₀₉	—	4	8	no	—
57	C₂×C₄×C₁₀₉	—	4	16	no	—
58	C₂×C₄×C₁₀₉	C₂×C₄²	1	8	yes	—
59	C₂×C₄×C₁₀₉	C₂×C₄²	1	2	yes	—
60	C₂×C₄×C₁₀₉	#117	1	16	yes	—
61	C₂×C₄×C₁₀₉	#117	1	4	yes	—
62	C₂×C₄×C₁₀₉	C₂³×C₄	1	2	yes	—
63	C₂×C₄×C₁₀₉	C₂×#41	1	4	yes	—
64	C₈×C₁₀₉	—	2	16	no	—
65	C₂²×C₁₀₉	—	2	4	no	—
66	C₂²×C₁₀₉	—	2	8	no	—
67	C₂²×C₁₀₉	#376	1	4	yes	—
68	C₂²×C₁₀₉	C₂²×C₄²	1	1	yes	characteristic
69	C₂²×C₁₀₉	C₂×#117	1	2	yes	—
70	C₂²×C₁₀₉	#568	1	4	yes	—
71	C₄×C₁₀₉	—	2	8	no	—
72	C₄×C₁₀₉	—	4	8	no	—
73	C₄×C₁₀₉	#376	1	4	yes	—
74	C₄×C₁₀₉	C₂×#117	1	4	yes	—
75	#20365	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×C₁₀₉	—	2	4	no	—
77	C₂×C₁₀₉	C₂×#376	1	1	yes	characteristic
78	C₂×C₁₀₉	#2873	1	2	yes	—
79	C₂×C₁₀₉	#2921	1	4	yes	—
80	C₂×C₄×M₁₆	C₂×C₁₀₉	1	2	yes	—
81	C₂×#376	C₂×C₁₀₉	1	1	yes	characteristic
82	#1746	C₂×C₁₀₉	1	8	yes	complemented
83	#1728	C₂×C₁₀₉	1	4	yes	—
84	C₁₀₉	#20365	1	1	yes	characteristic, direct product factor, Sylow
85	C₂²×M₁₆	C₂²×C₁₀₉	1	4	yes	—
86	C₂²×C₄²	C₂²×C₁₀₉	1	2	yes	—
87	C₂²×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
88	C₄×M₁₆	C₄×C₁₀₉	1	16	yes	complemented
89	C₂×#97	C₂²×C₁₀₉	1	4	yes	—
90	C₂×C₄×C₈	C₂²×C₁₀₉	1	4	yes	—
91	#376	C₄×C₁₀₉	1	4	yes	—
92	#376	C₂²×C₁₀₉	1	4	yes	—
93	C₂×#95	C₄×C₁₀₉	1	4	yes	—
94	#336	C₂²×C₁₀₉	1	16	yes	—
95	C₂³×C₄	C₂×C₄×C₁₀₉	1	4	yes	—
96	C₂³×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
97	C₂³×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
98	C₂×M₁₆	—	2	16	no	—
99	C₂×M₁₆	C₂×C₄×C₁₀₉	1	16	yes	—
100	C₂²×C₈	C₂³×C₁₀₉	1	8	yes	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	C₂×C₄×C₁₀₉	1	4	yes	—
103	C₂×C₄²	C₂×C₄×C₁₀₉	1	2	yes	—
104	C₂×C₄²	C₂×C₄×C₁₀₉	1	8	yes	—
105	C₂×C₄²	C₂³×C₁₀₉	1	4	yes	—
106	C₂×C₄²	C₂³×C₁₀₉	1	2	yes	—
107	#97	C₂×C₄×C₁₀₉	1	16	yes	complemented
108	C₄×C₈	C₂×C₄×C₁₀₉	1	16	yes	complemented
109	#95	C₂×C₄×C₁₀₉	1	16	yes	—
110	C₂⁴	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	4	4	no	—
113	C₂²×C₄	—	4	8	no	—
114	C₂²×C₄	C₄²×C₁₀₉	1	8	yes	—
115	C₂²×C₄	C₂²×C₄×C₁₀₉	1	8	yes	—
116	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
117	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
118	C₂²×C₄	C₂²×C₄×C₁₀₉	1	4	yes	—
119	C₂²×C₄	#41×C₁₀₉	1	8	yes	—
120	C₂²×C₄	C₂⁴×C₁₀₉	1	1	yes	characteristic
121	M₁₆	—	2	32	no	—
122	C₂×C₈	—	2	16	no	—
123	C₂×C₈	C₂²×C₄×C₁₀₉	1	16	yes	—
124	C₄²	—	2	8	no	—
125	C₄²	—	2	8	no	—
126	C₄²	C₄²×C₁₀₉	1	8	yes	—
127	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
128	C₄²	C₂²×C₄×C₁₀₉	1	4	yes	—
129	C₂³	—	2	4	no	—
130	C₂³	C₂×C₄²×C₁₀₉	1	2	yes	—
131	C₂³	#117×C₁₀₉	1	4	yes	—
132	C₂³	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
133	C₂×C₄	—	2	8	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	16	no	—
136	C₂×C₄	—	4	8	no	—
137	C₂×C₄	—	4	16	no	—
138	C₂×C₄	C₂×C₄²×C₁₀₉	1	8	yes	—
139	C₂×C₄	C₂×C₄²×C₁₀₉	1	2	yes	—
140	C₂×C₄	#117×C₁₀₉	1	16	yes	—
141	C₂×C₄	#117×C₁₀₉	1	4	yes	—
142	C₂×C₄	C₂³×C₄×C₁₀₉	1	2	yes	—
143	C₂×C₄	C₂×#41×C₁₀₉	1	4	yes	—
144	C₈	—	2	16	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	#376×C₁₀₉	1	4	yes	—
148	C₂²	C₂²×C₄²×C₁₀₉	1	1	yes	characteristic
149	C₂²	C₂×#117×C₁₀₉	1	2	yes	—
150	C₂²	#568×C₁₀₉	1	4	yes	—
151	C₄	—	2	8	no	—
152	C₄	—	4	8	no	—
153	C₄	#376×C₁₀₉	1	4	yes	—
154	C₄	C₂×#117×C₁₀₉	1	4	yes	—
155	C₂	—	2	4	no	—
156	C₂	C₂×#376×C₁₀₉	1	1	yes	characteristic
157	C₂	C₁₀₉×#2873	1	2	yes	—
158	C₂	C₁₀₉×#2921	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₀₉	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₀₉
socle	C₂³×C₁₀₉	C₂³×C₄
prime radical	C₂⁴×C₁₀₉	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₁₀₉×#1746	C₂	semidirect
8. C₄×M₁₆×C₁₀₉	C₄	semidirect
27. #97×C₁₀₉	C₂×C₄	semidirect
28. C₄×C₈×C₁₀₉	C₂×C₄	semidirect
75. #20365	C₁₀₉	direct (N × H)
82. #1746	C₂×C₁₀₉	semidirect
84. C₁₀₉	#20365	direct (N × H)
88. C₄×M₁₆	C₄×C₁₀₉	semidirect
107. #97	C₂×C₄×C₁₀₉	semidirect
108. C₄×C₈	C₂×C₄×C₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×M₁₆×C₁₀₉	2
C₂×#376×C₁₀₉	2
C₁₀₉×#1746	2
C₁₀₉×#1728	2
#20365	109

Subgroups — C109×#20365 #42138359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#20365 #42138359